Finding number of states using Myhill Nerode:
We have 7 possibilities for a string in $\Sigma ^{*}$. They are the strings whose last 2 symbols are 00, 01, 10, 11, 1, 0, $\epsilon$. Among them 00 and 11 belong to final state and other belong to Non final.
Let us find out who are distinguishable from each other
00 distinguishable from 11 by 1
00 distinguishable from 10 by $\epsilon$
00 distinguishable from 01 by $\epsilon$
00 distinguishable from 1 by $\epsilon$
00 distinguishable from 0 by $\epsilon$
00 distinguishable from $\epsilon$ by $\epsilon$
11 distinguishable from 10 by $\epsilon$
11 distinguishable from 01 by $\epsilon$
11 distinguishable from 1 by $\epsilon$
11 distinguishable from 0 by $\epsilon$
11 distinguishable from $\epsilon$ by $\epsilon$
01 distinguishable from 10 by 1
01 distinguishable from 1 by nothing so they are equivalent. Hence 01$\equiv$1.
01 distinguishable from 0 by 1
01 distinguishable from $\epsilon$ by 1
10 distinguishable from 1 by 1
10 distinguishable from 0 by nothing so they are equivalent.Hence 10$\equiv$0.
So we have 5 equivalence classes. {00}, {11}, {01,1}, {10,0}, {$\epsilon$}.
Hence the number of states in minimal DFA is 5.